Posted by Anonymous on Thursday, January 19, 2012 at 10:33pm.
the sides of a triangle have lengths 4x+1, 2x+1 and 6x1. if the lengths of the longest side is 6x1, what values of x make the triangle obtuse?

geometry  Reiny, Thursday, January 19, 2012 at 11:52pm
(6x1)^2 > (4x+1)^2 + (2x+1)^2 ,
36x^2  12x + 1 = 16x^2 + 8x + 1 + 4x^2 + 4x + 1
6x^2  24x 1 > 0
suppose we solve the corresponding equation
6x^2  24x 1 = 0
x = (24 ± √600)/12
=(24 ± 10√6)/24
= (12 ± 5√6)/12 = appr 2.0206 or a negative
but clearly 6x1 > 0
x > 1/6
so it is obtuse for x > 2.0206
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